Publications

Foreword: Special issue of JSC on the occasion of MEGA 2019 2022
Bernardi, A.; D'Andrea, C.; Theobald, T., "Foreword: Special issue of JSC on the occasion of MEGA 2019" in JOURNAL OF SYMBOLIC COMPUTATION, v. 2022, n. 109 (2022), p. 199201.  URL: https://www.sciencedirect.com/science/article/pii/S0747717120300560 .  DOI: 10.1016/j.jsc.2020.07.001
2022 journal paper

Strict inclusions of high rank loci 2022
Ballico, Edoardo; Bernardi, Alessandra; Ventura, Emanuele, "Strict inclusions of high rank loci" in JOURNAL OF SYMBOLIC COMPUTATION, v. 2022, n. 109 (2022), p. 238249.  URL: https://www.sciencedirect.com/science/article/pii/S0747717120300596 .  DOI: 10.1016/j.jsc.2020.07.004
For a given projective variety $X$, the high rank loci are the closures ofthe sets of points whose $X$rank is higher than the generic one. We showexamples of strict inclusion between two consecutive high rank loci. Our firstexample is for the Veronese surface of plane quartics. Although Piene hadalready shown an example when $X$ is a curve, we construct infinitely manycurves in $mathbb P^4$ for which such strict inclusion appears. For spacecurves, we give two criteria to check whether the locus of points of maximalrank 3 is finite (possibly empty).
2022 journal paper

SKEWSYMMETRIC TENSOR DECOMPOSITION 2021
Esteban Arrondo, Enrique; Bernardi, Alessandra; Macias Marques, Pedro; Mourrain, Bernardi, "SKEWSYMMETRIC TENSOR DECOMPOSITION" in COMMUNICATIONS IN CONTEMPORARY MATHEMATICS, v. 23, n. 2 (2021), p. 195006101195006129.  URL: https://www.worldscientific.com/doi/10.1142/S0219199719500615 .  DOI: 10.1142/S0219199719500615
2021 journal paper

Geometric conditions for strict submultiplicativity of rank and border rank 2021
Ballico, Edoardo; Bernardi, Alessandra; Gesmundo, Fulvio; Oneto, Alessandro; Ventura, Emanuele, "Geometric conditions for strict submultiplicativity of rank and border rank" in ANNALI DI MATEMATICA PURA ED APPLICATA, v. 200, n. 1 (2021), p. 187210.  URL: https://link.springer.com/article/10.1007/s10231020009916 .  DOI: 10.1007/s10231020009916
2021 journal paper

High order singular value decomposition for plant diversity estimation 2021
Bernardi, Alessandra; Iannacito, Martina; Rocchini, Duccio, "High order singular value decomposition for plant diversity estimation" in BOLLETTINO DELLA UNIONE MATEMATICA ITALIANA, v. 2021, n. 14(4) (2021), p. 557591.  URL: https://doi.org/10.1007/s4057402100300w .  DOI: 10.1007/s4057402100300w
2021 journal paper

rasterdiv—An Information Theory tailored R package for measuring ecosystem heterogeneity from space: To the origin and back 2021
Rocchini, Duccio; Thouverai, Elisa; Marcantonio, Matteo; Iannacito, Martina; Da Re, Daniele; Torresani, Michele; Bacaro, Giovanni; Bazzichetto, Manuele; Bernardi, Alessandra; Foody, Giles M.; Furrer, Reinhard; Kleijn, David; Larsen, Stefano; Lenoir, Jonathan; Malavasi, Marco; Marchetto, Elisa; Messori, Filippo; Montaghi, Alessandro; Moudrý, Vítězslav; Naimi, Babak; Ricotta, Carlo; Rossini, Micol; Santi, Francesco; Santos, Maria J.; Schaepman, Michael E.; Schneider, Fabian D.; Schuh, Leila; Silvestri, Sonia; Ŝímová, Petra; Skidmore, Andrew K.; Tattoni, Clara; Tordoni, Enrico; Vicario, Saverio; Zannini, Piero; Wegmann, Martin, "rasterdiv—An Information Theory tailored R package for measuring ecosystem heterogeneity from space: To the origin and back" in METHODS IN ECOLOGY AND EVOLUTION, v. 2021, 12, n. 6 (2021), p. 10931102.  URL: https://besjournals.onlinelibrary.wiley.com/doi/10.1111/2041210X.13583 .  DOI: 10.1111/2041210X.13583
2021 journal paper

Identifiability of Rank3 Tensors 2021
Ballico, Edoardo; Bernardi, Alessandra; Santarsiero, Pierpaola, "Identifiability of Rank3 Tensors" in MEDITERRANEAN JOURNAL OF MATHEMATICS, v. 18, n. 4 (2021).  DOI: 10.1007/s00009021017884
2021 journal paper

Waring, tangential and cactus decompositions 2020
Bernardi, Alessandra; Taufer, Daniele, "Waring, tangential and cactus decompositions" in JOURNAL DE MATHÉMATIQUES PURES ET APPLIQUÉES, v. 143, (2020), p. 130.  URL: https://www.sciencedirect.com/science/article/abs/pii/S0021782420301203 .  DOI: 10.1016/j.matpur.2020.07.003
2020 journal paper

On the ranks of the third secant variety of SegreVeronese embeddings 2019
Ballico, Edoardo; Bernardi, Alessandra, "On the ranks of the third secant variety of SegreVeronese embeddings" in LINEAR & MULTILINEAR ALGEBRA, v. 2019, 67, n. 3 (2019), p. 583597.  URL: https://www.tandfonline.com/doi/abs/10.1080/03081087.2018.1430117 .  DOI: 10.1080/03081087.2018.1430117
2019 journal paper

A note on the cactus rank for SegreVeronese varieties 2019
Ballico, Edoardo; Bernardi, Alessandra; Gesmundo, Fulvio, "A note on the cactus rank for SegreVeronese varieties" in JOURNAL OF ALGEBRA, v. 526, (2019), p. 611.  URL: https://www.sciencedirect.com/science/article/pii/S002186931930078X .  DOI: 10.1016/j.jalgebra.2019.01.027
We give an upper bound for the cactus rank of any multihomogeneous polynomial.
2019 journal paper

On the partially symmetric rank of tensor products of Wstates and other symmetric tensors 2019
Ballico, Edoardo; Bernardi, Alessandra; Christandl, Matthias; Gesmundo, Fulvio, "On the partially symmetric rank of tensor products of Wstates and other symmetric tensors" in ATTI DELLA ACCADEMIA NAZIONALE DEI LINCEI. RENDICONTI LINCEI. MATEMATICA E APPLICAZIONI, v. 30, n. 1 (2019), p. 93124.  URL: http://www.emsph.org/journals/journal.php?jrn=rlm .  DOI: 10.4171/RLM/837
Given tensors $T$ and $T'$ of order $k$ and $k'$ respectively, the tensor product $T otimes T'$ is a tensor of order $k+k'$. It was recently shown that the tensor rank can be strictly submultiplicative under this operation ([ChristandlJensenZuiddam]). We study this phenomenon for symmetric tensors where additional techniques from algebraic geometry are available. The tensor product of symmetric tensors results in a partially symmetric tensor and our results amount to bounds on the partially symmetric rank. Following motivations from algebraic complexity theory and quantum information theory, we focus on the socalled emph{$W$states}, namely monomials of the form $x^{d1}y$, and on products of such. In particular, we prove that the partially symmetric rank of $x^{d_1 1}y ootimes x^{d_k1} y$ is at most $2^{k1}(d_1+ cdots +d_k)$.
2019 journal paper

Introduction 2019
Ballico, Edoardo; Bernardi, Alessandra; Carusotto, Iacopo; Mazzucchi, Sonia; Moretti, Valter, "Introduction" in "Quantum Physics and Geometry", by Edoardo Ballico, Alessandra Bernardi, Iacopo Carusotto, Sonia Mazzucchi, Valter Moretti, edited by Edoardo Ballico, Alessandra Bernardi, Iacopo Carusotto, Sonia Mazzucchi, Valter Moretti, M. Joseph Landsberg, Davide Pastorello, Bassano Vacchini, Frédéric Holweck, Luca Chiantini, F. M. Ciaglia, A. Ibort, G. Marmo, Switzerland: Springer, Cham, 2019, p. 49.  URL: https://link.springer.com/chapter/10.1007/9783030061227_1 .  DOI: 10.1007/9783030061227_1
2019 part of book

Quantum Physics and Geometry 2019
Ballico, Edoardo; Bernardi, Alessandra; Carusotto, Iacopo; Mazzucchi, Sonia; Moretti, Valter (edited by), "Quantum Physics and Geometry", by Edoardo Ballico, Alessandra Bernardi, Iacopo Carusotto, Sonia Mazzucchi, Valter Moretti, Luca Chiantini, Frédéric Holweck, M. Joseph Landsberg, F.M. Ciaglia, Alberto Ibort, Giuseppe Marmo, Davide Pastorello, Bassano Vacchini, Switzerland: Springer, 2019, 172 p.  (LECTURE NOTES OF THE UNIONE MATEMATICA ITALIANA).  ISBN: 9783030061210.  URL: https://www.springer.com/gp/book/9783030061210 .  DOI: 10.1007/9783030061227
2019 book

On polynomials with given Hilbert function and applications 2018
Bernardi, Alessandra; Jelisiejew, Joachim; Macias Marques, Pedro; Ranestad, Kristian, "On polynomials with given Hilbert function and applications" in COLLECTANEA MATHEMATICA, v. 2018, n. Volume 69, Issue 1 (2018), p. 3964.  URL: https://link.springer.com/article/10.1007/s1334801601902 .  DOI: 10.1007/s1334801601902
Using Macaulay’s correspondence we study the family of Artinian Gorenstein local algebras with fixed symmetric Hilbert function decomposition. As an application we give a new lower bound for the dimension of cactus varieties of the third Veronese embedding. We discuss the case of cubic surfaces, where interesting phenomena occur.
2018 journal paper

Typical and Admissible ranks over fields 2018
Ballico, Edoardo; Bernardi, Alessandra, "Typical and Admissible ranks over fields" in RENDICONTI DEL CIRCOLO MATEMATICO DI PALERMO, v. 67, n. 1 (2018), p. 115128.  URL: https://link.springer.com/article/10.1007/s1221501702995?wt_mc=Internal.Event.1.SEM.ArticleAuthorAssignedToIssue .  DOI: 10.1007/s1221501702995
Let $X(mathbb{R})$ be a geometrically connected variety defined over $mathbb{R}$ such that the set of all its complex points $X(mathbb{C})$ is nondegenerate. We introduce the notion of emph{admissible rank} of a point $P$ with respect to $X$ to be the minimal cardinality of a set $Ssubset X(mathbb{C})$ of points such that $S$ spans $P$ and $S$ is stable under conjugation. Any set evincing the admissible rank can be equipped with a emph{label} keeping track of the number of its complex and real points. We show that, in the case of generic identifiability, there is an open dense euclidean subset of points with certain admissible rank for any possible label. Moreover we show that if $X$ is a rational normal curve then there always exists a label for the generic element. We present two examples in which either the label doesn't exist or the admissible rank is strictly bigger than the usual complex rank.
2018 journal paper

Singularities of plane rational curves via projections 2018
Bernardi, Alessandra; Gimigliano, Alessandro; Idà, Monica, "Singularities of plane rational curves via projections" in JOURNAL OF SYMBOLIC COMPUTATION, v. 86, n. May–June (2018), p. 189214.  URL: http://dx.doi.org/10.1016/j.jsc.2017.05.003 .  DOI: 10.1016/j.jsc.2017.05.003
We consider the parameterization ${mayhbf{f}}=(f_0:,f_1:f_2)$ of a plane rational curve $C$ of degree $n$, and we study the singularities of $C$ via such parameterization. We use the projection from the rational normal curve $C_nsubsetmathbb{P}^n$ to $C$ and its interplay with the secant varieties to $C_n$. IN particular, we define via $mathbf{f}$ certain 0dimensioal schemes $X_ksubset mathbb{P}^k$, $2leq k leq (n1)$, which encode all information on the singularities of multiplicity $geq k$ of $C$ (e.g. using $X_2$ we can give a criterion to determine whether $C$ is a cuspidal curve or has only ordinary singularities). We give a series of algorithms which allow one to obtain information about the singularities from such schemes.
2018 journal paper

On real typical ranks 2018
Grigoriy, Blekherman; Ottaviani, Giorgio; Bernardi, Alessandra, "On real typical ranks" in BOLLETTINO DELLA UNIONE MATEMATICA ITALIANA, v. 2018, 11, n. 3 (2018), p. 293307.  URL: https://link.springer.com/article/10.1007/s4057401701340 .  DOI: 10.1007/s4057401701340
2018 journal paper

A new class of nonidentifiable skew symmetric tensors 2018
Bernardi, Alessandra; Vanzo, Davide, "A new class of nonidentifiable skew symmetric tensors" in ANNALI DI MATEMATICA PURA ED APPLICATA, v. 2018, n. Volume 197, Issue 5 (2018), p. 14991510.  URL: https://link.springer.com/article/10.1007/s102310180734z?wt_mc=alerts.TOCjournals&utm_source=toc&utm_medium=email&utm_campaign=toc_10231_197_5 .  DOI: 10.1007/s102310180734z
We prove that the generic element of the fifth secant variety $sigma_5(Gr(mathbb{P}^2,mathbb{P}^9)) subset mathbb{P}(igwedge^3 mathbb{C}^{10})$ of the Grassmannian of planes of $mathbb{P}^9$ has exactly two decompositions as a sum of five projective classes of decomposable skewsymmetric tensors. {We show that this, {together with $Gr(mathbb{P}^3, mathbb{P}^8)$, is the only nonidentifiable case} among the nondefective secant varieties $sigma_s(Gr(mathbb{P}^k, mathbb{P}^n))$ for any $n<14$. In the same range for $n$, we classify all the weakly defective and all tangentially weakly defective secant varieties of any Grassmannians.} We also show that the dual variety $(sigma_3(Gr(mathbb{P}^2,mathbb{P}^7)))^{ee}$ of the variety of 3secant planes of the Grassmannian of $mathbb{P}^2subset mathbb{P}^7$ is $sigma_2(Gr(mathbb{P}^2,mathbb{P}^7))$ the variety of bisecant lines of the same Grassmannian. The proof of this last fact has a very interesting physical interpretation in terms of measurement of the entanglement of a system of 3 identical fermions, the state of each of them belonging to a 8th dimensional ``Hilbert'' space.
2018 journal paper

On the dimension of contact loci and the identifiability of tensors 2018
Ballico, Edoardo; Bernardi, Alessandra; Chiantini, Luca, "On the dimension of contact loci and the identifiability of tensors" in ARKIV FÖR MATEMATIK, v. 56, n. 2 (2018), p. 265283.  URL: http://www.intlpress.com/site/pub/pages/journals/items/arkiv/content/vols/0056/0002/a004/index.html .  DOI: 10.4310/ARKIV.2018.v56.n2.a4
Let X⊂ℙr be an integral and nondegenerate variety. Set n:=dim(X). We prove that if the (k+n−1)secant variety of X has (the expected) dimension (k+n−1)(n+1)−1
2018 journal paper

Bounds on the tensor rank 2018
Ballico, Edoardo; Bernardi, Alessandra; Chiantini, Luca; Guardo, Elena, "Bounds on the tensor rank" in ANNALI DI MATEMATICA PURA ED APPLICATA, v. 2018, 197, n. 6 (2018), p. 17711785.  URL: http://link.springer.com/article/10.1007/s1023101807486 .  DOI: 10.1007/s1023101807486
2018 journal paper

The Hitchhiker Guide to: Secant Varieties and Tensor Decomposition 2018
Bernardi, Alessandra; Carlini, Enrico; Virginia Catalisano, Maria; Gimigliano, Alessandro; Oneto, Alessandro, "The Hitchhiker Guide to: Secant Varieties and Tensor Decomposition" in MATHEMATICS, v. 6, n. 12 (2018), p. 3140131486.  URL: https://www.mdpi.com/22277390/6/12/314/htm .  DOI: 10.3390/math6120314
We consider here the problem, which is quite classical in Algebraic geometry, of studying the secant varieties of a projective variety X. The case we concentrate on is when X is a Veronese variety, a Grassmannian or a Segre variety. Not only these varieties are among the ones that have been most classically studied, but a strong motivation in taking them into consideration is the fact that they parameterize, respectively, symmetric, skewsymmetric and general tensors, which are decomposable, and their secant varieties give a stratification of tensors via tensor rank. We collect here most of the known results and the open problems on this fascinating subject.
2018 journal paper

Curvilinear schemes and maximum rank of forms 2017
Ballico, E; Bernardi, Alessandra, "Curvilinear schemes and maximum rank of forms" in LE MATEMATICHE, v. 72, n. 1 (2017), p. 137144.  URL: https://lematematiche.dmi.unict.it/index.php/lematematiche/article/view/1360/1015 .  DOI: 10.4418/2017.72.1.10
We define the \emph{curvilinear rank} of a degree $d$ form $P$ in $n+1$ variables as the minimum length of a curvilinear scheme, contained in the $d$th Veronese embedding of $\mathbb{P}^n$, whose span contains the projective class of $P$. Then, we give a bound for rank of any homogenous polynomial, in dependance on its curvilinear rank.
2017 journal paper

A uniqueness result on the decompositions of a bihomogeneous polynomial 2017
Ballico, Edoardo; Bernardi, Alessandra, "A uniqueness result on the decompositions of a bihomogeneous polynomial" in LINEAR & MULTILINEAR ALGEBRA, v. 2017, 65, n. 4 (2017), p. 677698.  URL: https://www.tandfonline.com/doi/full/10.1080/03081087.2016.1202182 .  DOI: 10.1080/03081087.2016.1202182
In the first part of this paper we give a precise description of all the minimal decompositions of any bihomogeneous polynomial $p$ (i.e. a partially symmetric tensor of $S^{d_1}V_1\otimes S^{d_2}V_2$ where $V_1,V_2$ are two complex, finite dimensional vector spaces) if its rank with respect to the SegreVeronese variety $S_{d_1,d_2}(V_1,V_2)$ is at most $\min \{d_1,d_2\}$. Such a polynomial may not have a unique minimal decomposition as $p=\sum_{i=1}^r\lambda_i p_i$ with $p_i\in S_{d_1,d_2}(V_1,V_2)$ and $\lambda_i$ coefficients, but we can show that there exist unique $p_1, \ldots , p_{r'}$, $p_{1}', \ldots , p_{r''}'\in S_{d_1,d_2}(V_1,V_2) $, two unique linear forms $l\in V_1^*$, $l'\in V_2^*$, and two unique bivariate polynomials $q\in S^{d_2}V_2^*$ and $q'\in S^{d_1}V_1^*$ such that either $p=\sum_{i=1}^{r'} \lambda_i p_i+l^{d_1}q $ or $ p= \sum_{i=1}^{r''}\lambda'_i p_i'+l'^{d_2}q'$, ($\lambda_i, \lambda'_i$ being appropriate coefficients). In the second part of the paper we focus on the tangential variety of the SegreVeronese varieties. We compute the rank of their tensors (that is valid also in the case of SegreVeronese of more factors) and we describe the structure of the decompositions of the elements in the tangential variety of the twofactors SegreVeronese varieties.
2017 journal paper

Tensor decomposition and homotopy continuation 2017
Bernardi, Alessandra; Noah, Daleo; Jonathan, Hauenstein; Bernard, Mourrain, "Tensor decomposition and homotopy continuation" in DIFFERENTIAL GEOMETRY AND ITS APPLICATIONS, v. 2017, n. 55 (2017), p. 78105.  URL: https://www.sciencedirect.com/science/article/pii/S0926224517301055?via=ihub .  DOI: 10.1016/j.difgeo.2017.07.009
2017 journal paper

On parameterizations of plane rational curves and their syzygies 2016
Bernardi, Alessandra; Gimigliano, A.; Idà, M., "On parameterizations of plane rational curves and their syzygies" in MATHEMATISCHE NACHRICHTEN, v. 289, n. 56 (2016), p. 537545.  URL: http://www3.interscience.wiley.com/journal/60500208/home .  DOI: 10.1002/mana.201500264
Let $C$ be a plane rational curve of degree $d$ and $p: ilde C ightarrow C$ its normalization. We are interested in the {it splitting type} $(a,b)$ of $C$, where $mathcal{O}_{mathbb{P}^1}(ad)oplus mathcal{O}_{mathbb{P}^1}(bd)$ gives the syzigies of the ideal $(f_0,f_1,f_2)subset K[s,t]$, and , $(f_0,f_1,f_2)$ is a parameterization of $C$. We want to describe in which cases $(a,b)=(k,dk)$ ($2kleq d)$, via a geometric description; namely we show that $(a,b)=(k,dk)$ if and only if $C$ is the projection of a rational curve on a rational normal surface in $PP^{k+1}$.
2016 journal paper